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D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"37fbb6779085362e906641ffecd7faa2df894ab56f50dcbe41b89bfe6f15c2c2","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-55697ddae4d22d60","role":"sequence","name":"Triangle read by rows: T(n,k) = n!*(n+k-1)!/((n-k)!*(n-1)!*(k!)^2) for 0 <= k <= n, with T(0,0) = 1.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:  1;\n   2:  2,    1;\n   3:  5,   10,     3;\n   4: 10,   45,    48,     10;\n   5: 17,  136,   300,    220,     35;\n   6: 26,  325,  1200,   1750,    980,     126;\n   7: 37,  666,  3675,   8680,   9450,    4284,     462;\n   8: 50, 1225,  9408,  32340,  55860,   48510,   18480,    1716;\n   9: 65, 2080, 21168,  99120, 244860,  332640,  240240,   78936,   6435;\n  10: 82, 3321, 43200, 263340, 873180, 1675674, 1873872, 1158300, 334620, 24310;\n","recurrence_sentence":"The row polynomials satisfy (4*n^4-44*n^3+167*n^2-253*n+126)*P(n,x) = -(4*n^4-52*n^3+239*n^2-455*n+300)*P(n-4,x) + (16*n^4-200*n^3+888*n^2-1640*n+1056+(32*n^4-400*n^3+1784*n^2-3324*n+2160)*x)*P(n-3,x) - (24*n^4-288*n^3+1226*n^2-2172*n+1350+(64*n^4-768*n^3+3280*n^2-5856*n+3672)*x+(64*n^4-768*n^3+3296*n^2-5952*n+3780)*x^2)*P(n-2,x) + (16*n^4-184*n^3+744*n^2-1240*n+720+(32*n^4-368*n^3+1496*n^2-2532*n+1512)*x)*P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 2 + x, P(3,x) = 5 + 10*x + 3*x^2, P(4,x) = 10 + 45*x + 48*x^2 + 10*x^3.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2 + x;\nP[3, x_] := 5 + 10*x + 3*x^2;\nP[4, x_] := 10 + 45*x + 48*x^2 + 10*x^3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(4*n^4 - 52*n^3 + 239*n^2 - 455*n + 300)*P[n - 4, x] + (16*n^4 - 200*n^3 + 888*n^2 - 1640*n + 1056 + (32*n^4 - 400*n^3 + 1784*n^2 - 3324*n + 2160)*x)*P[n - 3, x] - (24*n^4 - 288*n^3 + 1226*n^2 - 2172*n + 1350 + (64*n^4 - 768*n^3 + 3280*n^2 - 5856*n + 3672)*x + (64*n^4 - 768*n^3 + 3296*n^2 - 5952*n + 3780)*x^2)*P[n - 2, x] + (16*n^4 - 184*n^3 + 744*n^2 - 1240*n + 720 + (32*n^4 - 368*n^3 + 1496*n^2 - 2532*n + 1512)*x)*P[n - 1, x])/(4*n^4 - 44*n^3 + 167*n^2 - 253*n + 126)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); 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